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In many areas of science, the study of network structures and properties can yield useful and profound results, and this is particularly true of the study of so- cial networks, where a substantial literature has been developed [Scott 2012]. In the course of studying complex systems, networks are typically an effective way of representing relational data. Examples can be found among social systems (friend and acquaintance networks), ecological systems (food webs), biochemi- cal systems, technological systems (e.g.: power networks, the internet), logistic systems, communication systems and the list goes on. Networks of interest are often very large, with thousands or millions of nodes and millions or billions of edges (links between nodes). Typically there is extra meta-data associated with network nodes or edges [Fortunato et al. 2013]. Extracting useful information and insights from such data is an important and non-trivial task.

2.3.1

Identifying Communities

Identifying communities or clusters in networks — groups of nodes that are well connected internally and less connected externally — is an important and active research topic. Community structures can provide direct insights into hidden relationships (e.g.: functional groups in protein networks [Ou-Yang et al. 2014]) and give a higher level view of large and complex networks that may be more amenable to human interpretation [Fortunato 2012].

Much work on community detection in networks has used a model where communities are disjoint, so that no community overlap is possible [Fortunato 2010]. Here the problem is posed as finding a partition of the network, where each partition is considered a community. A range approaches to this formulation exist with a corresponding rich literature [Tang and Liu 2010]. Let us consider an undirected network (or equivalently “graph”) consisting of a collection of “nodes”5 and the “edges” that connect them6_{. For example, individual Twitter users could} be the nodes, and mutual Twitter follower relations the edges.

One approach to non-overlapping community detection of note, and which has been widely used in the research literature, is often referred to as “modularity clustering”. This approach considers the difference between the number of edges connecting pairs of network nodes and the expected number of edges given the degree distribution. This quantity, summed over node pairs within a group of nodes is known as the groups “modularity”. If m is the total number of edges,

di is the degree of node i (the number of edges connected to node i), Aij is the

network’s adjacency matrix (Aij = 1 if there is an edge between nodesi and j, 0

otherwise) and C a collection of nodes, the modularityQC of C can be written:

Qc= 1 2m X i,j∈C;i6=j (Aij −didj/2m) (2.1)

Modularity can be either positive or negative, with positive values indicat- ing the possible presence of community structure. Modularity clustering seeks to partition the network such that the sum of the partition modularities is max- imised [Newman 2006].

A second approach of note draws on an information theoretic view of the prob- lem, seeking to optimise the information about the original network contained in the cluster representation [Rosvall and Bergstrom 2007]. This approach was found to perform as well as modularity clustering on evenly distributed clusters, but outperformed modularity clustering (and other approaches) on data where the community structure in the network was uneven, with some small and some large communities.

In many cases, and, in particular, with social networks, communities can be expected to overlap, and recently a substantial body of work has been devoted

5_{The term “vertex” is often used also.}

6_{For simplicity, we consider only undirected unweighed edges and only one possible edge}

to overlapping community detection [Xie et al. 2013]. Of particular note is the mixed membership stochastic block model [Airoldi et al. 2009], a Bayesian model for overlapping community detection, and the efficient inference algorithm for this model developed in [Gopalan and Blei 2013]. This model and inference technique can be said to be state of the art, showing notable improvements on various standard data sets and being applicable to massive networks in reasonable time.

2.3.2

Twitter Networks

Network metrics of Twitter data have revealed varying and quite different charac- teristics. For example, a study of early Twitter data (2009) from Singapore found 72% reciprocity (friend links that “follow back”) [Weng et al. 2010] whereas an- other 2009 study that obtained a nearly complete snapshot of the Twitter network found on average only 10% reciprocity [Cha et al. 2010].

Kwak et al. [Kwak et al. 2010] attempted to crawl the entire Twitter follower graph. They collected user profiles by snowball sampling [Biernacki and Wal- dorf 1981] from famous blogger Perez Hilton7 also adding users who mentioned trending topics8 during the period of data collection (approximately one month), claiming to have crawled “the entire Twitter-sphere” — not an entirely unrea- sonable assertion, since any Twitter account they missed can be said to not have contributed to public discourse through either mentioning popular topics or con- nections in the follower network. They present a number of general statistics on the friend/follower network and other user, tweet and network characteristics. Of particular note are their findings that a user’s retweet count does not correlate with the number of followers nor PageRank [Page et al. 1999] in the follower net- work; there is a non-power-law follower distribution, short effective diameter, and low reciprocity in the follower network. All of these are unexpected in the light of known characteristics of human social networks [Newman and Park 2003].

Huberman et al. [Huberman et al. 2008] investigated the relation between the Twitter follower network and the mention network. When one posts a tweet, it is possible to include another user’s user name preceded by an “_@” symbol, in which case the other user is alerted to the tweet. This is known as “mentioning” and was taken as a closer indication of true social networks than the follower network. They found that real social networks (approximated by the mention network) are

7_{Perez Hilton had over a million followers at the time of the study.}

8_{Trending topics are hash tags, keywords and phrases identified by Twitter as “trending”,}

subgraphs of the follower network. In another example, [Jurgens 2013] found that the location of nearby users in the mention network to be useful in inferring the location of users for which no direct location information is available.

Retweets, when a twitter user reproduces a tweet verbatim (possibly with some small addition of their own) provide another opportunity to extract a user network from a collection of tweets. A retweet network is typically thought of as a network of information flows. A number of studies have utilised retweet networks. For example in [Conover et al. 2011], community structures in a retweet network were found to be good predictors of political alignment.

Several other Twitter studies described in the following section (Section2.3.3) looked at aspects of network dynamics.

2.3.3

Dynamics of Social Networks

Social network dynamics has been established as playing an important role in coordinated action. In a controlled study by Rand et al. [Rand et al. 2011], people preferentially added social ties to cooperative people, and broke them with uncooperative people. A substantial body of work exists on link prediction in networks (social or otherwise) [L¨u and Zhou 2011; Wang et al. 2014]. Almost all of this work focusses on the prediction of new future links or missing links, with no attention given to predicting the dissolution of links or presence of spurious links [Wang et al. 2014], though two recent studies defy this pattern.

Kwak et.al. [Kwak et al. 2011] studied “unfollow” behaviour of 1.2 million Korean speaking Twitter users. They took daily snapshots of the user’s friend lists and attempted to identify factors leading to unfollow events — when a user has been following another users tweets, but decides to no longer follow them. Factors that were found to be significant were reciprocity in follow relationships (users unfollowed users who did not follow them), the duration of the relationship (users unfriended users they had not been following for long), the followee’s infor- mativeness (if a user has retweeted or favorited followee’s tweets) and overlap of the user and followee’s friend and follow lists. They also conducted a survey of 22 Twitter users to determine their motivations for unfollowing, identifying frequent tweeters, tweeters of uninteresting topics and tweeters of mundane details of their lives to be significant motivators.

Another study that looked at unfollow behaviour [Xu et al. 2013] also looked at the Korean Twitter network. They took four snapshots of nearly 700,000 Korean Twitter users friend and follower lists. Focussing on ordinary users in tightly

knitted groups they found that relational properties such as mutual following and followers in common reduce the likelihood of unfollowing. They found that unfollowing tends to be reciprocal — if someone unfollows you, you are more likely to unfollow them in return. They found no evidence that common interests and informativeness of interactions impacted unfollow behaviour. Their work suggests that there may be many diverse types of Twitter user groups where the impact of relational and informational factors may differ.