What are the steps of influencing a BGP route ?

• Local Preference: Local preference is used to choose the outbound external BGP path. Local preference is sent to all internal BGP routers in the autonomous system, and the path with the highest local preference is preferred. The default Local Preference value is100.
• Weight: Weight is a Cisco-defined attribute that is local to a router. The weight attribute is not advertised to neighboring routers.  If the router learns about more than one route to the same destination, the route with the highest weight will be preferred

BGP path selection is dependent on an algorithm which has elements ranking in order.For example first influencer is the Weight Metric where BGP prefers the path with the highest metric but Weight is locally significant.I'll post a detailed informative article of it.

BGP assigns the first valid path as the current best path. BGP then compares the best path with the next path in the list, until BGP reaches the end of the list of valid paths. This list provides the rules that are used to determine the best path:

1. Prefer the path with the highest WEIGHT.

   Note:  WEIGHT is a Cisco-specific parameter. It is local to the router on which it is configured.

2. Prefer the path with the highest LOCAL_PREF.

   Note: A path without LOCAL_PREF is considered to have had the value set with the bgp default local-preferencecommand, or to have a value of100 by default.

3. Prefer the path that was locally originated via a network or aggregate BGP subcommand or through redistribution from an IGP.

   Local paths that are sourced by the network or redistribute commands are preferred over local aggregates that are sourced by the aggregate-address command.

4. Prefer the path with the shortest AS_PATH.

   Note: Be aware of these items:

   • This step is skipped if you have configured the bgp bestpath as-path ignore command.

   • An AS_SET counts as1, no matter how many ASs are in the set.

   • The AS_CONFED_SEQUENCE and AS_CONFED_SET are not included in the AS_PATH length.

5. Prefer the path with the lowest origin type.

   Note: IGP is lower than Exterior Gateway Protocol (EGP), and EGP is lower than INCOMPLETE.

6. Prefer the path with the lowest multi-exit discriminator (MED).

   Note: Be aware of these items:

   • This comparison only occurs if the first (the neighboring) AS is the same in the two paths. Any confederation sub-ASs are ignored.

     In other words, MEDs are compared only if the first AS in the AS_SEQUENCE is the same for multiple paths. Any preceding AS_CONFED_SEQUENCE is ignored.

   • If bgp always-compare-med is enabled, MEDs are compared for all paths.

     You must disable this option over the entire AS. Otherwise, routing loops can occur.

   • If bgp bestpath med-confed is enabled, MEDs are compared for all paths that consist only of AS_CONFED_SEQUENCE.

     These paths originated within the local confederation.

   • THE MED of paths that are received from a neighbor with a MED of4,294,967,295 is changed before insertion into the BGP table. The MED changes to to4,294,967,294.

   • THE MED of paths that are received from a neighbor with a MED of4,294,967,295 are considered valid and are inserted into BGP table with effect to Codes fixed for Cisco bug ID CSCef34800.

   • Paths received with no MED are assigned a MED of0, unless you have enabled bgp bestpath med missing-as-worst .

     If you have enabled bgp bestpath med missing-as-worst , the paths are assigned a MED of4,294,967,294.

     If you have enabled bgp bestpath med missing-as-worst , the paths are assigned a MED of4,294,967,295 with effect to Codes fixed for Cisco bug ID CSCef34800.

   • The bgp deterministic-med command can also influence this step.

     Refer to How BGP Routers Use the Multi-Exit Discriminator for Best Path Selection for a demonstration.

7. Prefer eBGP over iBGP paths.

   If bestpath is selected, go to Step9 (multipath).

   Note: Paths that contain AS_CONFED_SEQUENCE and AS_CONFED_SET are local to the confederation. Therefore, these paths are treated as internal paths. There is no distinction between Confederation External and Confederation Internal.

8. Prefer the path with the lowest IGP metric to the BGP next hop.

   Continue, even if bestpath is already selected.

9. Determine if multiple paths require installation in the routing table for BGP Multipath.

   Continue, if bestpath is not yet selected.

10. When both paths are external, prefer the path that was received first (the oldest one).

    This step minimizes route-flap because a newer path does not displace an older one, even if the newer path would be the preferred route based on the next decision criteria (Steps11,12, and13).

    Skip this step if any of these items is true:

    • You have enabled the bgp best path compare-routerid command.

      Note: Cisco IOS Software Releases12.0.11S,12.0.11SC,12.0.11S3,12.1.3,12.1.3AA,12.1.3.T, and12.1.3.E introduced this command.

    • The router ID is the same for multiple paths because the routes were received from the same router.

    • There is no current best path.

      The current best path can be lost when, for example, the neighbor that offers the path goes down.

11. Prefer the route that comes from the BGP router with the lowest router ID.

    The router ID is the highest IP address on the router, with preference given to loopback addresses. Also, you can use thebgp router-id command to manually set the router ID.

    Note: If a path contains route reflector (RR) attributes, the originator ID is substituted for the router ID in the path selection process.

12. If the originator or router ID is the same for multiple paths, prefer the path with the minimum cluster list length.

    This is only present in BGP RR environments. It allows clients to peer with RRs or clients in other clusters. In this scenario, the client must be aware of the RR-specific BGP attribute.

13. Prefer the path that comes from the lowest neighbor address.

    This address is the IP address that is used in the BGP neighbor configuration. The address corresponds to the remote peer that is used in the TCP connection with the local router.

1. Prefer the path with the highest WEIGHT.

   Note:  WEIGHT is a Cisco-specific parameter. It is local to the router on which it is configured.

2. Prefer the path with the highest LOCAL_PREF.

   Note: A path without LOCAL_PREF is considered to have had the value set with the bgp default local-preferencecommand, or to have a value of100 by default.

3. Prefer the path that was locally originated via a network or aggregate BGP subcommand or through redistribution from an IGP.

   Local paths that are sourced by the network or redistribute commands are preferred over local aggregates that are sourced by the aggregate-address command.

4. Prefer the path with the shortest AS_PATH.

   Note: Be aware of these items:

   • This step is skipped if you have configured the bgp bestpath as-path ignore command.

   • An AS_SET counts as1, no matter how many ASs are in the set.

   • The AS_CONFED_SEQUENCE and AS_CONFED_SET are not included in the AS_PATH length.

5. Prefer the path with the lowest origin type.

   Note: IGP is lower than Exterior Gateway Protocol (EGP), and EGP is lower than INCOMPLETE.

6. Prefer the path with the lowest multi-exit discriminator (MED).

   Note: Be aware of these items:

   • This comparison only occurs if the first (the neighboring) AS is the same in the two paths. Any confederation sub-ASs are ignored.

     In other words, MEDs are compared only if the first AS in the AS_SEQUENCE is the same for multiple paths. Any preceding AS_CONFED_SEQUENCE is ignored.

   • If bgp always-compare-med is enabled, MEDs are compared for all paths.

     You must disable this option over the entire AS. Otherwise, routing loops can occur.

   • If bgp bestpath med-confed is enabled, MEDs are compared for all paths that consist only of AS_CONFED_SEQUENCE.

     These paths originated within the local confederation.

   • THE MED of paths that are received from a neighbor with a MED of4,294,967,295 is changed before insertion into the BGP table. The MED changes to to4,294,967,294.

   • THE MED of paths that are received from a neighbor with a MED of4,294,967,295 are considered valid and are inserted into BGP table with effect to Codes fixed for Cisco bug ID CSCef34800.

   • Paths received with no MED are assigned a MED of0, unless you have enabled bgp bestpath med missing-as-worst .

     If you have enabled bgp bestpath med missing-as-worst , the paths are assigned a MED of4,294,967,294.

     If you have enabled bgp bestpath med missing-as-worst , the paths are assigned a MED of4,294,967,295 with effect to Codes fixed for Cisco bug ID CSCef34800.

   • The bgp deterministic-med command can also influence this step.

     Refer to How BGP Routers Use the Multi-Exit Discriminator for Best Path Selection for a demonstration.

7. Prefer eBGP over iBGP paths.

   If bestpath is selected, go to Step9 (multipath).

   Note: Paths that contain AS_CONFED_SEQUENCE and AS_CONFED_SET are local to the confederation. Therefore, these paths are treated as internal paths. There is no distinction between Confederation External and Confederation Internal.

8. Prefer the path with the lowest IGP metric to the BGP next hop.

   Continue, even if bestpath is already selected.

9. Determine if multiple paths require installation in the routing table for BGP Multipath.

   Continue, if bestpath is not yet selected.

The mostly used attributes are local preference and autonomous system (AS) number. Local preference is used to  update all the BGP routers to use the path which has to be taken to the internet, the higher the Local preference the better the path. AS number is advertised in BGP updates to the ISP routers to influence ISP routers to take the path to corporates internal network, the lower the AS number the better the path.

Hope I answered your question in a precise manner.