A Discrete Bat algorithm for the community detection ...

A Discrete Bat algorithm for the community detection problem Eslam Ali Hassan1,3 , Ahmed Ibrahem Hafez2,3 , Aboul Ella Hassanien1,3 , Aly A. Fahmy1 1 Faculty of Computers and Information, Cairo University, Egypt [email protected],[email protected],[email protected] 2 Faculty of Computer and Information, Minia University, Egypt [email protected] 3 Scientific Research Group in Egypt (SRGE), http://www.egyptscience.net

Abstract. Community detection in networks has raised an important research topic in recent years. The problem of detecting communities can be modeled as an optimization problem where a quality objective function that captures the intuition of a community as a set of nodes with better internal connectivity than external connectivity is selected to be optimized. In this work the Bat algorithm was used as an optimization algorithm to solve the community detection problem. Bat algorithm is a new Nature-inspired metaheuristic algorithm that proved its good performance in a variety of applications. However, the algorithm performance is influenced directly by the quality function used in the optimization process. Experiments on real life networks show the ability of the Bat algorithm to successfully discover an optimized community structure based on the quality function used and also demonstrate the limitations of the BA when applied to the community detection problem. Keywords: Community detection, Community Structure, Social Networks, Bat algorithm, Nature-inspired algorithms

1

Introduction

A social network can be modeled as a graph composed of nodes that are connected by one or more specific types of relationships, such as friendship, values, work. The purpose of community detection in networks is to recognize the communities by only using the information embedded in the network topology. Many methods have been developed for the community detection problem. These methods use tools and techniques from different disciplines like physics, chemistry, applied mathematics, and computer and social sciences [1]. One of the main interests in social network analysis is discovering community structure. A Community is a group of nodes that are tightly connected to each other and loosely connected with other nodes. Community detection is the process of network clustering into similar groups or clusters. Community detection has many applications including visualization, detecting communities of special interest,

2

Authors Suppressed Due to Excessive Length

realization of the network structure [2], etc [3]. One of the recent techniques in community detection is Girvan-Newman (GN) algorithm [4]. Girvan-Newman is a divisive method that uses the edge betweenness as a measure to discover the boundaries of communities. This measure detects the edges between communities through counting the number of shortest paths between two specific nodes that passes through a special edge or node. Later on Girvan and Newman introduced a new technique called Modularity [5]. Modularity measures the community strength of a partition of the network, where high Modularity means strong community structure that has dense inter-connections between the community’s nodes, Thus the problem of community detection can be viewed as a Modularity Maximization problem. Finding the optimal Modularity is an NPComplete problem, many heuristic search algorithms have been investigated to solve this problem such as genetic algorithm (GA), simulated annealing, artificial bee colony optimization (ABC) [1]. The remainder of this paper is organized as follows. In Section 2 we formulate the community detection problem and show the objective function used in the research. In Section 3 we illustrate The Basic Bat algorithm. In Section 4 we illustrate our proposed algorithm. Section 5 shows our experimental result on real life social networks. Finally we provide conclusions in section 6.

2

The community detection Problem

A social network can be viewed as a graph G = (V , E ), where V is a set of nodes, and E is a set of edges that connect two nodes of V . A community structure S in a network is a solution to the problem which is a set of communities of nodes that have a bigger density of edges among the nodes and a smaller density of edges between different sub-groups. The problem of detecting m communities in a network, where the number m is unknown can be formalized as finding a clustering of the nodes in m subsets that can best satisfy a given quality measure of communities F(S ). The problem can be formalized as an optimization problem where one usually wants to optimize the given fitness measure F(S ) [6]. The objective function has a significant role in the optimization process; it’s the “steering wheel” in the process that leads to good solutions. A lot of objective functions have been introduced to capture the intuition of communities, and there does not exist a direct method to compare those objective functions based on their definitions [7–9]. Network Modularity [5] is one of the most used quality measure of communities in the literature. Modularity measures the number of within-community edges relative to a null model of a random graph with the same degree distribution. We use community Modularity as our objective function in the Bat algorithm.

3

Bat Algorithm

The Bat Algorithm (BA), that was presented by Xin-She Yang [10], is a new meta-heuristic optimization algorithm derived by simulating the echolocation

A Discrete Bat algorithm for the community detection problem

3

system of bats. It is becoming a promising method because of its good performances in solving many problems [11]. Biological inspiration In nature, the echolocation behavior of bats used for hunting and navigation, where a bat emits ultrasound pulses to the surrounding environment, then listens back to the echoes in order to locate and identify preys and obstacles as shown in Figure 1. Each bat in the swarm can find the more nutritious area by individual search or moving forward towards a more nutritious location in the swarm. The basic idea of the BA is to imitate the echolocation behavior of bats with local search of bat individual for achieving the global optimum. At the beginning of the search, each individual bat emits pulses with low frequency but with greater loudness in order to cover bigger area in the search space, later on when a bat approaches from its prey, it increases its pulse emission rate and decreases the pulse loudness, this frequency/loudness adjustment process can control the balance between the exploration and the exploitation operations of the algorithm. 3.1

Bat Movements Description

Applying the algorithm to the optimization problem, generally a ‘bat’ represents an individual in a population. The environment in which the artificial bat lives is mainly the solution space and the states of other artificial bats. Its next movement depends on its current state, velocity and its environmental state (including the quality and state of the best bats in the swarm). Let the state vector of an artificial bat x composed of n variables such that x = (x1 , x2 , . . . , xn ) , and the velocity vector of artificial bat v composed of n variables such that v = (v1 , v2 , . . . , vn ) , fi is the current frequency that moves between fmin and fmax , vit is a new velocity selected according to equation 2 and xti a new state selected as in equation 3. Where β ∈ [0, 1] is a random vector drawn from a uniform distribution, x∗ is the current best global solution that is determined after selecting the best solutions between all the n bats in the current population, Because the product λi fi is the increase in velocity, then we can either use fi (or λi ) to modify the velocity while fixing the other parameter according to the problem type. fi = fmin + (fmax − fmin ) ∗ β

(1)

vit = vit−1 + (xti − x∗ ) ∗ fi

(2)

xti = xit−1 + vit

(3)

We chose fmin = 0 and fmax = 1, and draw initial frequency of each virtual bat from a uniform distribution f0 ∈ [fmin , fmax ]. Performing a local search, one of the best solutions in the current population is selected randomly, then a new solution is produced using 4, Where  ∈ [0, 1] is

4

Authors Suppressed Due to Excessive Length

Fig. 1: Shows the echolocation behavior of an Artificial Bat approaching its prey

random number, and At = hAti i is the average loudness of all bats in the current population. xnew = xold + At

3.2

(4)

Pulse emission rate and Loudness description

Moreover, the pulse emission rate r and the loudness A of each virtual bat must be updated when a new population generated, As soon as the bat found its prey, its loudness decreases and pulse emission rate increases according to equations 5 and 6 At+1 = αAt i

(5)

rit+1 = ri0 [1 − exp(−γt)]

(6)

Where α and γ are constants, after performing some experiments we chose α = 0.99, γ = 0.02, ri0 = 0.01 and A0i = 0.99 , Also we can notice that Ati ← 0, rit ← ri0 as the virtual bat gets closer to its prey t ← ∞, which is rational since a bat has just reached its prey and stops emitting any sound. This update process will only occur when a new better solution generated, which imply that this bat moves towards an optimal solution.

4

Proposed Algorithm

It is not possible to directly apply the Bat algorithm to the problem of community detection. First the algorithm should be redesigned. In the current section we illustrate the modified Bat algorithm so it can be applied to the community detection problem. The algorithm is outlined in listing 1.

A Discrete Bat algorithm for the community detection problem

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

5

Input: A Network G = (V , E ) Output: Community membership assignments for network’s nodes Initialize the parameters: pulse rates ri , loudness Ai , Swarm size np, the maximum number of iterations M ax Iterations Randomly initialize each artificial bat in the swarm with a random possible solution as its current state, the velocity of each bat, and calculate its fitness repeat Generate new solutions by adjusting frequency, and updating velocities and locations/solutions [equations 2 to 4 and the modified equations 7 and 8] if rand > ri then Select a solution among the best solutions Generate a local solution around the selected best solution end if rand < Ai and f(xi ) < f(x∗ )) then Accept the new solutions Increase ri and decrease Ai end t←t+1 until t > M ax Iterations; return the best solution achieved

Algorithm 1: Bat algorithm

4.1

Parameters redesign

The first step of modifying a BA for solving the problem of community detection, To provide an appropriate scheme of representation of an individual artificial bat in a population. The locus-based adjacency encoding scheme [12, 13] is selected to represent the solution. In that representation, each artificial bat state x composed of n elements (x1 , x2 , . . . , xn ) and each element can take a value j in the range [1 .. n]. A value j set to the ith element is translated as an association between node i and node j. Which means that, in the community structure detected, nodes i and j will exist in the same community. Normally the bat swarm will contain np artificial bats (AB). The bat current state represents a solution in the search space and the fitness value of the solution represents the amount of food resource at that location. The food condensation in the location of an artificial bat is formulated as yi = f (xi ) , Where yi is the fitness function value associated with xi calculated using Modularity quality measure. 4.2

Bat Movement

Bat movement are described in equations 1, 2 and 3. In a discrete problem representation such equations can not be applied directly. First the difference operator between to bat position will be calculated using equation 7; where g(xi ) is the group assignment of node i in the solution represented by bat position x.  1 if g(xi ) 6= g(x∗i ) di = (xi − x∗i ) = (7) −1 if g(xi ) = g(x∗i )

6

Authors Suppressed Due to Excessive Length

Fig. 2: Shows how a new solution is generated using  and the old solution X when the average loudness At = 0.6. Now equation 3 will be considered as uniform crossover between (x, x∗ ) using velocity vector v as the mixing ratio controller, so the new position value will be updated using equation 8.  ∗ xi if vi ≥ 1 (8) xnew = i xi otherwise 4.3

Local Search

In order to create a new solution using equation 4 , First  should be converted into a vector of size n of uniformly distributed random numbers between 0 and 1, then perform the generation process by selecting a new element from one of the neighbors of the ith node when i > At , otherwise keep the old neighbor as illustrated in Figure 2. 4.4

Current Global Best

The Basic Bat algorithm uses x∗ the current global best solution in updating all VB in the swarm, which will lead to moving all bats to the same location. In optimization problems this will might lead to trap the algorithm in a local optima. To overcome this problem instead of using one global best, we select η top bats according to their fitness value, then for each VB update a randomly selected VB from the current η top best is used. η is set 10% of the population size np by trail and error. Since the algorithm employs stochastic process to find optimal solution, it may converge to different solutions (non-deterministic). It is therefore not uncommon to run the algorithm multiple T times i.e. number of restarts, starting with initial different population in each iteration (chosen randomly) and then returning the best solution found across all runs according to the objective function used in the optimization process.

5

Experimental results

In the section we tested our algorithm on a real life social networks for which a ground truth communities partitions is known. To compare the accuracy of the

A Discrete Bat algorithm for the community detection problem

7

resulting community structures; we used Normalized Mutual Information (NMI) [14] to measure the similarity between the true community structures and the detected ones. Since Modularity is a popular community quality measure used extensively in community detection, we used it as a quality measure for the result community structure of all other objectives. We applied our algorithm on the following social networks datasets :– The Zachary Karate Club: which was first analyzed in [15], contains the community structure of a karate club. The network consists of 34 nodes. Due to a conflict between the club president and the karate instructor, the network is divided into two approximately equal groups. The network consists of 34 nodes and 78 edges. – The Bottlenose Dolphin network: was compiled by Lusseau [16] and is based on observations over a period of seven years of the behaviour of 62 bottlenose dolphins living in Doubtful Sound, New Zealand. The network split naturally into two large groups. – American College football network: [4] represent football games between American colleges during a regular season in Fall 2000, nodes in the graph represent teams and edges represent regular-season games between the two teams they connect. Games are more frequent between members of the same conference than between members of different conferences, the network is divided into 12 conferences. – Facebook Dataset: Leskovec [17] collects some data for the Facebook website -10 ego networks-. The data was collected from survey participants using a Facebook application [7]. The ago network consist of a user’s –the ego node– friends and their connections to each other. The 10 Facebook ego networks from [17] are combined into one big network. The result network is undirected network which contain 3959 nodes and 84243 edges. Despite there is no clear community structure for the network, a ground truth structure was suggested in [18]. For each dataset; we applied the Bat algorithm 10 restarts and calculated the NMI and Modularity value of the best solution selected. This process was repeated 10 times and average NMI and average Modularity is reported. The Bat algorithm was applied with the following parameters values; number of VB in the population np = 100 and the maximum number of iterations M ax Iterations = 100. Table 1 summarizes the average NMI and Modularity for the result obtained using the Bat algorithm. We observed that the result for the each network is better that its ground truth in term of Modularity. The detected community structures for each network is visualized in Fig.3. Figure 3a shows a visualization of the result for the Zachary network. The original division of the network is indicated by the vertical line and the detected structure is indicated by the nodes’ colors. As we can observe that the detected community structure contains 4 communities with a high Modularity value = 0.4696 in which the top level is similar to the original division of the network, however in the detected structure each group is farther divided into two groups.

8

Authors Suppressed Due to Excessive Length

Table 1: Modularity Result Zachary Karate Bottlenose Dolphin College Football Facebook Modularity 0.4696 0.5498 0.612 0.753 Ground Truth 0.4213 0.395 0.563 0.7234 NMI

0.6873

0.5867

0.84786

0.67374

Thus the NMI of comparing the detected structure with the ground truth of Zachary network is 0.6873 i.e. 68% of similarity between the two structures. Despite that the NMI value is somehow low, it is not a major judgmental criteria of result for two reasons. First the major quality criteria is Modularity value, since the algorithm is optimizing Modularity objective. Second the know ground truth of the network from the original study [15] might not be the optimal one and it is possible that there is a more modular structure that the ground truth. Figure 3b visualizes the result for the Bottlenose Dolphin network. As before the original division of the network is indicated by the curved line and the detected structure is indicated by the nodes’ colors. The original division of the Bottlenose Dolphin network is divided into 2 groups with Modularity value of 0.395. The detected community structure by the Bat algorithm is divided into 5 groups and has a higher Modularity value of 0.5498. Regarding the College football network; the original division of the network into conferences is highlighted in Fig.3c; only edges between nodes from the same group are shown and nodes’ color refer to which groups they belong to. From Fig.3c we can observe that some nodes never played any match with other nodes from their group. The detected community structure for this network is shown in Fig.3d which contains 10 communities with a more modular structure than the one shown in Fig.3c. Figure 3e shows a visualization of the largest 12 communities from the detected community structure of the Facebook dataset. As we can observe that each group show a dense connection between nodes from the same group and spare or low interactions between nodes from different groups except for a few nodes that has a large edge degree cross different groups such as nodes {1750, 476 and 294} which make group membership assignment a hard process for the algorithm. 5.1

Comparison Analysis

Now we compare the result obtained by Bat algorithm with other methods in the literature which are AFSA community detecton [19], Infomap [20], Fast greedy [21], Label propagation [22], Multi-level or Louvain [23], Walktrap [24] and leading Eignvector [25]. Each method is run 10 times for each dataset and the average NMI and Modularity of the result community structure is reported. Figure.4 summarize the NMI and Modularity values for all methods. As we can observe that in term of NMI; Bat algorithm produces a good result compared

A Discrete Bat algorithm for the community detection problem

(a) Result for Zachary Karate network. 80 31 56 30 20 36

2

34

110

26 38 90 46

106 104

86

100 72 55

13

19 35

44 39

83

43

51

29

74

89

66

24 94 17 105 10 5

50 84

4

108

99

114 97 96

88 63

6 85 82 53 41

86

100 72 55

13

19 35 37

44 39

45 93

52 109 8 112 79 9 78 22 69 23

57

28 18 21

1

42

51

89

94

71 77

105

108

99

4

64 98

28 18 88 63 21

40 3 33 7 107 14 101 16 65 61 48

73 75

6 85 82 53 41

(c) Original division of the College Football

57 59

114

17 12 10

96

11 103

60

66

5

97

50 84

67 92

91

25

24 70 29

76

58 49 113 87

68 47 54

115 111 74

40 3 33 7 107 14 101 16 65 61 48

73 75

11 103

71 77

62 15 27

43

32

95 81 80 102 31 56 30 83 20 36

106 104

98

12

70

115 59

26 38 90 46

76

60

64

25

68 47 54

110

67 92 49 58 87 113111

45 93

1

42

2

34

81 91 37

52 109 8 112 79 9 78 22 69 23

(b) Result for Bottlenose Dolphin network.

62 15 27

32

95 102

9

(d) Result for the College Football

908 949910 920 868

897947901 967 1048946915944895 891 913

881

476

930926 942 918 890960 924 1089 957 945

1095

1122

1417

922

927

1086 1096 1123 898 956 925905865 943 939 909 899 1077

1159148018921655189514861475 1174

1075

1581

1617 1639 1173

965953

966 954

883 959921 951863 1074 1044 1091 893 1084

859875

896

929

1434

1783

434

916

851

1444

1118 1015 977 1424 998 1012 1119

1389

818

1120816

1496

1513

849 1390

3272

1585

1753 1915 1920

1977 1989 1927

1983 1987 1928 1981

1974 1978 1589 1590 1921 1979

1019

1972 1980 1975

1991

1990

523

1858

3177

3226

1140

1064

792

1263

1259 1446 12581535 9831007 1580 979 12621260 1431

1527 1269

999 1056

1664

10601267 822 1515 1261 12641398 1516 1533 1545 1518 1534 971 1508995

993 980

1004

1511

1510

1519 1532

1512 1268 1517

1720

1666

1057

443

1650

1681 1782

1266

1514 1509

3312 3387

1414

1062

380

1537

789

987

81 86 76 124 84 8378 80 77 119 85

810

125

29224 207

215

1441

1400

121

122

155

790

1826 1824 1825

226

209

8 55

69

40 195 197 113

132

444406

152

801

6

97103

68 37

94 63 35

11836 111 44

1

21

20 15 6034

53 109

73 52

43

64

14

24 104

93

115

4

42

98

271

1243

3274 3279 3353

3407

3388

3175

2782 2546

2299

2664

2747

2666

27742696

2489

2668

2667

2695 26772594

2548

764

2663

2693

2777

2814

2599

2694

2802

2813

640613667604

2537

2492

2595

2737

932

2799 2801

2614 2800 2618 2798

1070

737803

736

2874

2609

27682769

2772

2659

2883

2824

2387

2238

2443

2386

2687

2781

2587

2681

2589 1821

2701

1780 2394

2761

24132716 2407 2732 2811

2592

2408

486

2645

2422 2678

2699

1318

1401 13271372 1375

1320 1328 13161333 2385

3229

3269 3386

440

3163

1752

935

1498

29602946 2910 2913 2997 3029 2982 29343043 2957 29353063 2987 2893 3005 2955 3102 2992 29422976 30223068 3164 28963031 30423058 3109 2963 30983014 3018 1709 2916 2938 3010 3056 3006 3067 3000 30382951 2936 2900 3032 2967 2970 3181 2927 2928 30283015 2943 2969 2962 2984 2915 3012 2898 2925 30642953 3105 3008 2965 3001 2939 2972 2961 30922901 2890 3096 2903 2981 3026 29912980 3034 3055 2959 3065 3013 2973 3049 3027 28943019 3004 3089 2904 2920 3039 29072971 2999 3023 3093 2986 2918 3110 2899 3104 2929 3048 2919 3007 2930 3035 29953040 3030 2917 2994 3046 3103 3062 2998 2906 3057 2958 3069 2924 3053 2948 2947 2891 3020 30912905 2945 3060 2993 29142956 3021 2944 29262941 2912 2931 3172

3228

1635

1437

3672 3744

3182 3095

3674

3675

3468 3712 35963426 37703777

3449 3448 3590 3636 36343425 384834443771

3612

3438

3702

3437 3594 35653775 3471

3759 35593728 3757

36243628

3566378234473844 3622 3810 3588 3589 3558 3776 3554 3895 3783 3615 3597 3595 344638413760 3632 1355 3555 362336103433 3723 37223727 3710 3818 2204 3765 3427 3568 3553 1904 3721 3704 3703 3635 3781 3724 3434 3616 3840 3846 37333849 3585 3847 359335081361 3732 36173619 3613 3621 3814 3625 3633 3845 3809 1352 34693820 3719 3439 3764 3725 3720 3570 22023626 3627 3561 3908 3900 36113620 3819 3631 36303614 3466 1350 3430 3436 3706 3746 3774 3815 3586 3443 3668 3812 3591 3629 3904 3766 3707 3705 3711 3640 3580 3572 1353 3609 3578 3512

3837

1689

3811 3667 1344 1337 3860 3708 3587 3717 1363 1349 1360 3709 3644 3858 3582 34761347 13563569 3737 1346 1334 3574 1341 3859 3730 3718 3618 13573739 3567 3505 1340 1343 1359 3857 3772 3576 3891

3564 1362 3875

3581 1348

3753 3752 3738 3740 3741 3747

3536

3889

3743

3573 3465 3523

3642 36063550

3715

3504

3603

3520

3511

35283541 3641 3600 3513 3549 3607

3507 3545

3531 3524 3517 3599 3529 3532 3530 3604 3575 3645

3693

3533

3515 3543 3605

3510 3521 35183602 3544 3601

3716

3789 3793

3787 38013798

3790

3785

3797

3748

3749 3742

3516

3745

3869 3886

3880

3890

3868

3813

3577

3514

3643

3887

3639 3726

3583 1336 1351 1364 3473 1345 3571 35383534 3666 1338 3608 3474 13353467 3750 3638 3525 3546 1342 3598 3548 3579

3470 3526 3527 3537 35473522 13583472 3539 3503

3795 38003799

3349 3170

3133 3171 3100

3768

1308

2229

2209 2449 2236 2275

2278

2230 2237

3231

3173

1633

3788 37863794 3796 3792

3317

3249

1652

1502 1311 1315 1499

2733

3791 3802

3268

3270 3108

1253 1246 1252 1249

1248 1507

13211374 1506

3248 3400 3230 3251

3135

3180

1786

1255

2584 2208 2465 2211 2226 2274 2260 17812460 2452 2679 24002223 2251 2244 2462 1692 2248 2258 2454 2267 2257 2242 2447 2246 2231 2273 2217 2227 2253 2451 2390 2466 2444 22622206 2219 2222 22762256

2213

3385

547

1676

1254

1523

2680 2712

2564

3054 2911 3059 30502897 2974 2975 29503099 3045 3009 31062990 2988 29682909 2983 3036 3061 2985 2989 2932 3090 2952 2949 3097 3136 2977 3051 3024 2902 1700 3017 3134 3101 3052 3044 2895 29083002 2892 3003 2923 3066 2933 3107 3025 2940 3047 3037 2978 2937 3094

747

1604

1606 799 1677

1326

1371 1373 1325 1402 1319 1452 133213101313 808 1454 1314 1377

1317 1331 1323

1449

1828

2403

2397 2562

1384

3285

527

812

1787

1788

1183

1050

2821

2428 2418 239625722438 2700 1829 2575 2461 2569 1876 2734 2439 23802435 2634 2581 2578 2583 1823

2567

760

6751418 533

1185

1368

1385

1613

1550 1199 1685 1403 1764 1547 1309

1322

1453

1330 1051 1312 1378

1553

1439

1628

600 603 512556

1181 150116261455

811 1195

492

3155

2921 1802

794

1182

1609

1180

1369

564 1611

1053

2810 2812 2808 2809

2780

2426 2266 2632 1774 2463 2239 2580 2457 2269 2558 1836 1770 2404 237524021765 2405 2207 2234 2393 2252 2456 1779 1777 2265 2277 1579 2255 2264 2240 2379 2424 2247 15742423 2382 2270 2392 2212 2235 17712399 2464 24551562 2453 2221 22152727 2205 2218 1873 1566

2467 2585 2272 2216 2259 2254 22682450 2224 2249 2245 2243 2566 1878 15582389 2643

1827 1554

1599

1594 1598 1600 1595 1552 1784 14041751

2725 2806

2804

2648

1543

1376

491

3162 3399

3158 3152 3148

3147 3159 3151

3296

659

1433 687

734

934813 795 1184

690 804722 1068 707521 797

1419 721 68815491257 937

1450

1503

490

493

2684

2577 1573 2434 1556 23812469 2391 1571 2702 1776 2225 2401 2263 2214 2433 1766 1820 25912582

2647

809

1379

1831

2878

2442

2371 2743 2660 18742778 2746 2430 1578 2615 24592445 2674 2662 2417 2406 2412 2458 1772 2432 2427 23772671 1884 1778 1567 2233 1560 2395

2271

677 724

1380

2807

2232

2796

587 805 796

683

1524

271527142771

2565 237226201886

2767 260826032766 2751 26022384 2619 2571 2881 2738 2613 2626 2623 2383 2573 1775 27412640 2378 2628 2698 2605 2606 2776 1684 1565 2431 2673 2637 2415 2750 2374 2731 2672 2646 2752 2629 2588 2612 2429 2860 24202670 2376 2724 2411 2756 27082568 2758 25571832 2631 16952617 2723 1887 1563 2610 2762 2757 2669 2579 2370 424 2440 2763 2369 2555 188527832706 1769 2561 26012448 2649 26412744 183315721877 27492704 18222657 2437 2729 1575 2425 2650 26522421 2748 27601570 2398 2709 1875 2713 259027392785 2625 1568 2616 2661 1879 27302656 2436 225022412441 2707 2765 2388 2726 26112685 2604 2563 2468 2745 2638 2718 2446 2764 2586 2719 2228 2373 2416 2682 2409 2607 2742 2717 1682 1688 2822 2728 2419 18301561 25742721 1773 2740 1768 2560 2220 2635 2576

2683

3153 3160

3149 3115 3150 3154 3161 31573146

3250

569

553

1367

2784

3156

1721

787

793

2773

2644

294

731

1542

700 732

2496

1705

1115

814

1366

1432

2642

275

3380

786

1880

798

2261

1217

3266

349

347

1704

1546

1442

753 674

25702720

1814

3402

3295

3293

1798 1713

1250

1116

779

2410

1228

1717 1714 1223

3415 3414

1251

1399

425

2770

3241

3138

761

419

422

2703

252026002779

3247

304

1722 310 1229 12061699 1797 3168

3243 3131

684 776702 709 520 537726 766 499 615597 595655630 628704743 1067 577 551 581686 669 662 757 632579 571 528 582 591 696 651634 658717509 755762596 560 572689543501 534 720 506 576 739570 629 517 568 626 788660668 647730781701 627 524 590 751 697 513 633 725 676741 678672 652710 679 775 500 748 502 522593784616 745 738 525 518694 740 7 28 636 711 503 768 727 559716 657530601 705693586 544558555 546763718 715735 746535 496548 532511498 723519714 703 542 708 539 562 495554 692 729 585526 680 515778573 605541 557654 589 538 574 549 598 550602567 713682754765580552 695584

2472

2330

3166

214

2288 2797

2542

3165

3126

750

1188

2705

2845 2843 2841 2847

3183

1804

639645770 771 782 578933 497 625681650 649670 666 643638742 609566 8001117 620 712 637 646691780706 565 508656607 536 563510 802516

2333 24701767 2525 2554 2337

2516

3142

3176

1812

1069

2805

2551

3116 3128

3129

3316 3348 3346

599772531635529

936663

2736

2348 2875

617

611 774561 505777642 631588 661 606621 767769 1114719 614756749641733 592545 575 665685 653610 622 758 618 664 673 612 752 644 619 608 624759 1329

2597

2545 2366 2473 2527 2598 2311 25052534 2363 2368 2335 2351 2501 2298 2310 2352 25362521 2517 2315 2281 2512 2483 23602313 2307 2478 2541 2494 2327 2515 2303 2319 2329 2285 2486 2317 2480 2293 2332 2539 2535 2297 2553 23672308 252925142354 2530 2471 2323 2526 2487 23262476 2338 2550 2283 2314 2318 2479 2507 2347 2552 2350

26902491 2334 2364 2688 2488 2543 2676 2481 2342 2522 2320 2528 2538 2502 2655 2503 2593 2513 2509 2511 2531 2518 2341 2735 2331 2484 2485 2477 2559 2305 2654 2339 2653 2499 2510 23562324 2549 2834 2280 2547 2474 2346 24902722247525002497 2504 2292 2296 23552312 2349 2290 2343 2544

3179

3347

1818

3318

32943178

504 507540 698648

2665 2691 2692

2755

2753

2697 2533 2689

373

783

1799

1702 357

3123

3119

671

1807 1813

3167

277 3244 1210 3239

3207

31303280 3252

623

367

372 1231345 342 333325 328 283 1240 1222 299 293316 323 3306 34317111214 1809 1213 3235352 375 356 1718 317 3113 3169 3232 17011232 3233 3120 1227 336 3114 353 3259 1220 1237 285281 1805 3127 1816 1230 1202 3111 3260 324 3267 3265 1815 295 3245 3137 1201 3238 3264 354 18081716 319 1242 1696 3121 3144 3236 331 3139

3118 3304 3143 3122

3308

3174 3140

699

284

3278

3282

3281 3352

514

12351204 1211 330

318 12261800 309 1207 329 1706 1708 308 1221 376 1203 337 1197 362 351 306 300 314 12253263 3283 1233 344 1811 3124 3261 1212 12161205 350 365 3125 374 291

110

156 149

126159 228 229 232

227 230

146

1187

3254

3284 3188

3237

1698

3112

457

2294 2493 2325

3242

3309

66

62 130

133 158 127 134 223112114 148

154

181 139137 136 128 151

381

261

1817

1810 1803 312

1796

58 6717 47 3 48 105 56 38

71100 219

102 135 192 212180 147

161162

397

3256

3276 3132 3288

25 1970 95 140

1712 3551801287

3079340 335 276270 1710 368 326360278 322 1244 1236 296 377 741707 1697 364 338302366 3591703 307282 1215 1208 1806 1715 348 1239 1238 1245 3341723 332 3240 3234 32621218

459

150

807

3277 3311

16 18 267 13 65 45 96 1602 1011101 30

57 32

141 142 193 211 153 144 49

407 201

221210 145222

420

33 23 9 31 22 27 50 28 5 131

117

59

194 54

220143 216

268

92

12087 185 79 225106157

1724

303 1200 313 3117 3193320 36312241234 1219 272 3053076 3145 31411241 361 301 30822693713077 3081 1209

138

91

89 88 90

123

82

744

1686 1871

2344 2306 2495 2279 23002357 2506 2316 2322 2282 2519 2291 23212284 2286 2345 2358 2340 2302 2361 2289 23532304 2498 2523 2362 2287 2295 2540 2336 2508 2482 2309 2596 2365 2328 2359 2301 2532

3330

3199

3319 3202

3360

1587 1923 1911

19261982

1410

1020

3218

1582

1397

1631

843

1395

1059

1063

1763

8429701001

981 974 1528 8261445 840 1544

1602

583

1588 1988

1037 1065 1127

1016

1674

1128

1421

1426 1427 1627

3404 3289

33843275

1591

1058

1607

848

1265

994

1536 1370 99613861420834986973 969 821 11131009 975 1003 9921000 1121968978 997 976 1055 1388

1014 1392 1010 982 1391 837853 972 1416 1011 902

3379

1848 1870

1022 1458

1005

3324

3217

3305

3273 32123291 3331 3203 3219

3356 3204

1851

1869

1794

32253336

3298 3210 3214 3303 3339 3208 3314 3224 3227

1854

1846

1837 1842 1843 18641852 18681865 1855 1867

3222

3299 33013302

1866

1651

1440

1054

1256

1460

32213329

3325

3361 3363 33823334 3307 33223344 3383 3310 3397 3337 3286 32903326

3209 3381 3313 3323 3287 3321 3297 3328 3200 329232063327 3220 3341 3338 3211 3300 3396 3271 32233201

3315 3358

1669

1840

1839

1793 1792

3320 3342 321333573401 3335

3362 3364 3405

1933 1932

1754 454

455

478479

481 480

1028

1436 1673

938

1530

1603

1476 452

1934

1850 1905

1898

1916

1464

1665

1680 1795

1679

1017

1522 15391129

984

1890 1845

1835

1470

1478 1168 10244831461456 484

1018 1034 1039

1406 1130 1526

9851006

1192

4531396 1678

3345 3333

1857

18441618

1158 14661487

1139

791

1415 1540 1538 1061 1125 1126 14301505

1608

815

1634

1026

1131 1521

1520

1393

1013

825

836

839 845 1394 1186

1541

1500

873

1100 1108

436 1405

1658

1917

1040 831 1382 1030 1138 8201529 1409 1429 1412 1428

3390

1726 1913 1728 1853

1659

1653

1167 1668 14951175 1151 1620 1492 1494 1165 1491 1525 1148 1630 1160 1170 1667 1143

482 1023

1134 1038 1132 841838 1021

830

1411

1049

1908

17391849 1729

1755

1745 1169 1156 1162 11571465 1142 1036 1149 1624 14681463

1029

854 8291137

817 835

941

1008

1615

1457

1456 475

1098 1101 1191 1073 11931102 879 1085 1071 866

1111

1931 1930 1912

17401894

1897

1748

1747

1079 1080 857 10421083 1094 1082885 8701097 1093 1047

867 11101106

1909

1484 1733 1727 17431907 1731 1730 1660 1906 1178 1746 1741

1742 1657 1489 1144 18561738

11721469 1725 1736 1152 1474 1164 1177 1163 1176 1477 1661 1663173418411847 1638 1481 1154 1790 1472 1145 1171 1819 1744 1147

1136 847 10321027 8528551025 1383 1033 846850 1035 1616 1002 1041 833 828827 8241443 8191133 1381 1459 1531

880 856962860877882 864 858 884 886 1099 8761107 872 8781092 903 1407 861 10781103 10431190 1112 1105 1109 1104

1789

1914

1483

1737

888

958

963 1076

1081 1088871 1072 1087 887

1640

1153 1656 1479 1161 1471 1838 1654

1625 1732 1488 1619 1179 1662 1785 1735 1791 1150 1482 1637 1896 1749 1146 1485 14731155 1462

948

911

1893

1636 1621 1622 1623 1467 1166 1490

1090

964 950892 919955862 923 952 894

904 914 1046 940 869874931912 917 961 900906 889 907 1045928

3497

3888

3671

3803 3806 3882 3862 3484

3804 3483

3903

3778 3836 3490 3489 3779

3486 3477 34943495 3480 3488 3498 3881

3808

3502 3485 3478 3479 3493 3492 3835 36373482

3832 3501 3807 3834 3499 3902 3500 3670 3669

348734813878

3491 3496 38053659 38333873 3879 3647 381738163661

3655 3649 3650 3654 3663 3648 3662 36643651

3656 3657

3646 3665 3658 3653 3652 3660

3016 2922 3041 2966 2964 2979 2996 3011 2954 3033

(e) Result for Facebook dataset

Fig. 3: Visualizations of the result obtained by Bat Algorithm on each social network dataset.

10

Authors Suppressed Due to Excessive Length

to other methods as shown in Fig.4a. In term of Modularity; Bat algorithm is very competitive with other methods as shown in Fig.4b. For the small size data set we can observe the Bat algorithm produce a community structure with a high Modularity value compared to all other methods. Regarding the Facebook datasets; Bat algorithm failed to produce a result with high Modularity value compared to the other methods. 5.2

Discussion

As observed from our experimental result that the Bat algorithm performance is promising for small size networks, however for a large networks Bat algorithms performance is degraded compared to other CD algorithms. From our initial analysis, we found there is no much diversity in the VB swarm over the search space and the Bat algorithm does not explore a large region in the search space for the following reasons:

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6 0.5

Modularity

NMI

– All the population is moving toward one position (Current global best x∗ ). over iterations this will lead to all VBs will move/evolve to similar solutions. Despite that we overcome this problem using η top global best, it decreased its impact but did not eliminate it. – There is no operator/behavior that allow the VB to escape a local optima or jump/explore new random regions in the search space. For example in Genetic algorithm there exist a mutation operation that allow such behavior even a simple mutation in the current solution could cause a large diversity in the current population. Despite that Bat algorithm performance in other application [11] that are continues in nature is very promising, however for the community detection problem (discrete case) Bat algorithm performance has some limitations. – The local search and Bat difference operator that we proposed for the community structure Section 4.2 and 4.3 are not optimal and it is not clear if they are efficient in exploring the search space. It is possible for another design to cause a significant improvement to the algorithm performance.

0.4 0.3 0.2 0.1

0.5 0.4

0.3 0.2 0.1

0 Zachary Karate Bat

AFSA

Bottlenose Dolphin

Infomap

Fastgreedy

American College Football

Label Propagation

(a) NMI values.

Multilevel

Facebook Walktrap

0 Zachary Karate Bat

AFSA

Infomap

Bottlenose Dolphin Fastgreedy

American College Facebook Football Multilevel Walktrap Ground Truth

Label Propagation

(b) Modularity values.

Fig. 4: Average NMI and Modularity values of the result community structure obtained by each algorithm.

A Discrete Bat algorithm for the community detection problem

11

– Accepting criteria for new solution has some limitation. The basic Bat algorithm accept new solution only if it is better than the current global best. This may constrain the number of moves that a bat can perform.

6

Conclusions and Future Work

A discrete Bat algorithm is introduced for finding community structure in social network. The locus-based adjacency encoding scheme is applied to represent a community structure. The locus-based adjacency encoding scheme has a major advantage that it enables the algorithm to deduce the number of communities k without past knowledge about it. The BA uses Modularity Quality measure as the fitness function in the optimization process. Experimental results demonstrate that the performance of the bat algorithm is quite promising in terms of accuracy and successfully finds an optimized community structure based on the Modularity quality function for small size networks, however the performance is degraded for large size networks. BA algorithm produce good result for the small size network compared to other CD methods, however the result for the larger networks does not compete with other methods. In future work we are going to conduct an investigation of the discrete BA limitation introduced in Sec.5.2 to propose a new enhanced BA for the community detection problem and investigate other popular bio-inspired optimization algorithms, and apply it for the community detection problem, then conduct an analytical study between those methods to compare their performance.

References 1. Santo Fortunato. Community detection in graphs. Physics Reports, 486:75–174, 2010. 2. Ahmed S Ali, Ashraf S Hussien, Mohamed F Tolba, and Ahmed H Youssef. Visualization of large time-varying vector data. In Computer Science and Information Technology (ICCSIT), 2010 3rd IEEE International Conference on, volume 4, pages 210–215. IEEE, 2010. 3. Z. Masdarolomoor, R. Azmi, S. Aliakbary, and N. Riahi. Finding community structure in complex networks using parallel approach. In Embedded and Ubiquitous Computing (EUC), 2011 IFIP 9th International Conference on, pages 474–479, Oct 2011. 4. M. Girvan and M. E. J. Newman. Community structure in social and biological networks. Proceedings of the National Academy of Sciences, 99:7821–7826, 2002. 5. M.E.J Newman and M. Girvan. Finding and evaluating community structure in networks. Physics Rev. E, 69:026113, 2004. 6. Chuan Shi, Cha Zhong, Zhenyu Yan, Yanan Cai, and Bin Wu. A multi-objective approach for community detection in complex network. In IEEE Congress on Evolutionary Computation (CEC), pages 1–8. IEEE, 2010. 7. Jure Leskovec, Kevin J Lang, and Michael Mahoney. Empirical comparison of algorithms for network community detection. In Proceedings of the 19th international conference on World wide web, pages 631–640. ACM, 2010.

12

Authors Suppressed Due to Excessive Length

8. Chuan Shi, Philip S Yu, Yanan Cai, Zhenyu Yan, and Bin Wu. On selection of objective functions in multi-objective community detection. In Proceedings of the 20th ACM international conference on Information and knowledge management, pages 2301–2304. ACM, 2011. 9. Ahmed Ibrahem Hafez, Eiman Tamah Al-Shammari, Aboul ella Hassanien, and Aly A Fahmy. Genetic algorithms for multi-objective community detection in complex networks. In Social Networks: A Framework of Computational Intelligence, pages 145–171. Springer, 2014. 10. Xin-She Yang. A new metaheuristic bat-inspired algorithm. In JuanR. Gonzlez, DavidAlejandro Pelta, Carlos Cruz, Germn Terrazas, and Natalio Krasnogor, editors, Nature Inspired Cooperative Strategies for Optimization (NICSO 2010), volume 284 of Studies in Computational Intelligence, pages 65–74. Springer Berlin Heidelberg, 2010. 11. Xin-She Yang and Xingshi He. Bat algorithm: Literature review and applications. Int. J. Bio-Inspired Comput., 5(3):141–149, July 2013. 12. Chuan Shi, Cha Zhong, Zhenyu Yan, Yanan Cai, and Bin Wu. A new genetic algorithm for community detection. Complex Sciences, 5:1298–1309, 2009. 13. Clara Pizzuti. Community detection in social networks with genetic algorithms. pages 1137–1138, Atlanta, GA, USA, 2008. 14. L. Danon, A. Diaz-Guilera, J. Duch, and A. Arenas. Comparing community structure identification. Journal of Statistical Mechanics: Theory and Experiment, 9:09008, 2005. 15. W.W Zachary. An information flow model for conflict and fission in small groups. Journal of Anthropological Research, 33:452–473, 1977. 16. D. Lusseau. The emergent properties of dolphin social network. Proceedings of the Royal Society of London. Series B: Biological Sciences, 270:S186–S188, 2003. 17. Julian J. McAuley and Jure Leskovec. Learning to discover social circles in ego networks. pages 548–556, 2012. 18. Ahmed Ibrahem Hafez, Aboul Ella Hassanien, and Aly A Fahmy. Testing community detection algorithms: A closer look at datasets. In Social Networking, pages 85–99. Springer, 2014. 19. EslamAli Hassan, AhmedIbrahem Hafez, AboulElla Hassanien, and AlyA. Fahmy. Community detection algorithm based on artificial fish swarm optimization. In Intelligent Systems’2014, volume 323 of Advances in Intelligent Systems and Computing, pages 509–521. Springer International Publishing, 2015. 20. M. Rosvall, D. Axelsson, and C. T. Bergstrom. The map equation. The European Physical Journal Special Topics, 178(1):13–23, 2009. 21. Aaron Clauset, Mark EJ Newman, and Cristopher Moore. Finding community structure in very large networks. Physical review E, 70(6):066111, 2004. 22. Usha Nandini Raghavan, R´eka Albert, and Soundar Kumara. Near linear time algorithm to detect community structures in large-scale networks. Physical Review E, 76(3):036106, 2007. 23. Vincent D Blondel, Jean-Loup Guillaume, Renaud Lambiotte, and Etienne Lefebvre. Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment, 2008(10):P10008, 2008. 24. P. Pons and M. Latapy. Computing communities in large networks using random walks (long version). ArXiv Physics e-prints, 12 2005. 25. Mark EJ Newman. Finding community structure in networks using the eigenvectors of matrices. Physical review E, 74(3):036104, 2006.