Real world systems like biological, social, technological, infrastructural and many others can be modeled as networks. The field of network science aims to study these complex networks and understand their structure and dynamics. A common feature of networks across domains is the distribution of the degree of the nodes according to a power-law (scale invariance). As a consequence of this skewness, the high degree nodes dominate the properties of these networks. The rich-club phenomenon is observed when the high degree or the rich nodes of the network prefer to connect amongst themselves. In the first part, the thesis investigates the rich-club phenomenon in higher order neighborhoods of the network by providing an elegant quantification using a geodesic distance based approach. This quantification helped in identifying networks where the trend and intensity of the rich-club phenomenon is significantly different in higher order neighborhoods compared to the immediate neighbors. The thesis also proposes a quantification of the importance of the non-rich nodes in the communication structure of the rich nodes, and broadly classify networks into core-periphery or cellular. Further a lack of universality is noticed in the structure of the networks belonging to a particular domain. It has been observed in the previous literature that the rich club connectivity dominates assortativity, a measure quantifying the mixing patterns in complex networks. Thus, assortativity is biased. To overcome such drawbacks, in the second part of the thesis proposes a novel measure called regularity. The analytical bounds on regularity and formulation of regularity for different network models are provided. Along with this a measure to quantify the mixing patterns of the neighborhood of a node called local regularity is also defined. The analysis on real-world network based on local regularity and degree distribution shows presence of both type of network, uniformly and non-uniformly mixed across different regions. Further normalized regularity is proposed to quantify the extent of preferential mixing in networks discounting the effect of degree distribution.